Parabolic partial differential equation

Definition

To define the simplest kind of parabolic PDE, consider a real-valued function ${\displaystyle u(x,y)}$ of two independent real variables, ${\displaystyle x}$ and ${\displaystyle y}$. A second-order, linear, constant-coefficient PDE for ${\displaystyle u}$ takes the form

${\displaystyle A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+D{u}_x+E{u}_y+F=0,}$

where the subscripts denote the first- and second-order partial derivatives with respect to ${\displaystyle x}$ and ${\displaystyle y}$. The PDE is classified as parabolic if the coefficients of the principal part (i.e. the terms containing the second derivatives of ${\displaystyle u}$) satisfy the condition

${\displaystyle B^2-AC=0.}$

Usually ${\displaystyle x}$ represents one-dimensional position and ${\displaystyle y}$ represents time, and the PDE is solved subject to prescribed initial and boundary conditions. Equations with ${\displaystyle B^2-AC<0}$ are termed elliptic while those with ${\displaystyle B^2-AC>0}$ are hyperbolic. The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation ${\displaystyle A{x}^2+2Bxy+C{y}^2+Dx+Ey+F=0}$ to define a planar parabola.

The basic example of a parabolic PDE is the one-dimensional heat equation

${\displaystyle u_t=\alpha u_{xx},}$

where ${\displaystyle u(x,t)}$ is the temperature at position ${\displaystyle x}$ along a thin rod at time ${\displaystyle t}$ and ${\displaystyle \alpha }$ is a positive constant called the thermal diffusivity.

The heat equation says, roughly, that temperature at a given time and point rises or falls at a rate proportional to the difference between the temperature at that point and the average temperature near that point. The quantity ${\displaystyle u_{xx}}$ measures how far off the temperature is from satisfying the mean value property of harmonic functions.

The concept of a parabolic PDE can be generalized in several ways. For instance, the flow of heat through a material body is governed by the three-dimensional heat equation

${\displaystyle u_t=\alpha \mathrm{\Delta }u,}$

where

${\displaystyle \mathrm{\Delta }u:=\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{z}^2},}$

denotes the Laplace operator acting on ${\displaystyle u}$. This equation is the prototype of a multi-dimensional parabolic PDE.

Noting that ${\displaystyle -\mathrm{\Delta }}$ is an elliptic operator suggests a broader definition of a parabolic PDE:

${\displaystyle u_t=-Lu,}$

where ${\displaystyle L}$ is a second-order elliptic operator (implying that ${\displaystyle L}$ must be positive; a case where ${\displaystyle u_t=+Lu}$ is considered below).

A system of partial differential equations for a vector ${\displaystyle u}$ can also be parabolic. For example, such a system is hidden in an equation of the form

${\displaystyle \mathrm{\nabla }\cdot (a(x)\mathrm{\nabla }u(x))+b(x{)}^{\text{T}}\mathrm{\nabla }u(x)+cu(x)=f(x)}$

if the matrix-valued function ${\displaystyle a(x)}$ has a kernel of dimension 1.